Engel's theorem

In representation theory, a branch of mathematics, Engel's theorem is one of the basic theorems in the theory of Lie algebras; it asserts that for a Lie algebra two concepts of nilpotency are identical. A useful form of the theorem says that if a Lie algebra L of matrices consists of nilpotent matrices, then they can all be simultaneously brought to a strictly upper triangular form. The theorem is named after the mathematician Friedrich Engel, who sketched a proof of it in a letter to Wilhelm Killing dated 20 July 1890 (Hawkins 2000, p. 176). Engel's student K.A. Umlauf gave a complete proof in his 1891 dissertation, reprinted as (Umlauf 2010).

A linear operator T on a vector space V is defined to be nilpotent if there is a positive integer k such that T^k = 0. For example, any operator given by a matrix whose entries are zero on and below its diagonal, such as

is nilpotent. An element x of a Lie algebra L is ad-nilpotent if and only if the linear operator on L defined by

is nilpotent. Note that in the Lie algebra L(V) of linear operators on V, the identity operator I_V is ad-nilpotent (because ${\displaystyle \operatorname {ad} _{I_{V}}=0:L(V)\rightarrow L(V)}$) but is not a nilpotent operator.

A Lie algebra L is nilpotent if and only if the lower central series defined recursively by

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